In the theory of discrete time stochastic processes, a part of the mathematical theory of probability, the Doob decomposition theorem gives a unique decomposition of any submartingale as the sum of a martingale and an increasing predictable process. The theorem was proved by and is named for J. L. Doob.[1] The analogous theorem for continuous submartingales is the Doob–Meyer decomposition theorem.
Any submartingale Xn has a unique decomposition Xn = Mn + An where Mn is a martingale and An is a predictable, increasing process with A0 = 0.[2]